Additional regularity of optimal dual potentials and Monge-growth pairs

Establish the additional regularity required to obtain a regular optimal Monge-growth pairin particular, the $C^2$ regularity of a potential generating the pair and the associated $C^1$ controlfrom the presently available Lipschitz regularity theory for unbalanced optimal transport.

Background

The potential-generated parametrization requires a potential whose derivatives produce an admissible orientation-preserving diffeomorphism and a continuous growth factor. The exact-step analysis establishes only Lipschitz regularity of the relevant dual potential, whereas the neural approximation framework requires stronger regularity, including a regular potential in $C^2(\Omega)$.

The paper explains that the EulerLagrange equation is algebraic in the density and potential and does not itself provide an elliptic regularity gain. Consequently, stronger regularity for unbalanced optimal transport remains unresolved within the paper.